Network spot‐checking games: Theory and application to toll enforcing in transportation networks

Abstract

We introduce the class of spot‐checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero‐sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP‐hard. More generally, we prove that it is NP‐hard to compute a SSE in a polymatrix game, even if the game is pairwise zero‐sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 312–328 2015